

To evaluate \(\left\lbrace \!\left\lbrace \cdot \right\rbrace \!\right\rbrace _{\log }\)  in a numerically stable manner we follow the
approach outlined by [1]}. With these definitions, an
entropy conservative numerical flux for \(k = 1,2,3\)  is
\({\mathbf {\mathcal {A}}}_{k}(\mathbf {q}^{-}, \mathbf {x}^{-}, \mathbf {q}^{+}, \mathbf {x}^{+})&=\frac{1}{2}\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0\\0 & 1 & 0 & 0 & 0 & \delta _{1k} \hat{\rho }^{-}\\0 & 0 & 1 & 0 & 0 & \delta _{2k} \hat{\rho }^{-}\\0 & 0 & 0 & 1 & 0 & \delta _{3k} \hat{\rho }^{-}\\0 & 0 & 0 & 0 & 1 & 0\end{bmatrix},\\{\mathbf {\mathcal {H}}}_{k}(\mathbf {q}^{-}, \mathbf {x}^{-}, \mathbf {q}^{+}, \mathbf {x}^{+})&=\begin{bmatrix}{\left(\rho u_{k}\right)}^{*}\\{\left(\rho u_{k}\right)}^{*} \left\lbrace \!\left\lbrace u_{1}\right\rbrace \!\right\rbrace  + \delta _{1k} p^{*}\\{\left(\rho u_{k}\right)}^{*} \left\lbrace \!\left\lbrace u_{2}\right\rbrace \!\right\rbrace  + \delta _{2k} p^{*}\\{\left(\rho u_{k}\right)}^{*} \left\lbrace \!\left\lbrace u_{3}\right\rbrace \!\right\rbrace  + \delta _{3k} p^{*}\\e^{*} {\left(\rho u_{k}\right)}^{*} + \left\lbrace \!\left\lbrace u_{k}\right\rbrace \!\right\rbrace  p^{*}\\\phi \left(\mathbf {x}^{+}\right)\end{bmatrix}.\) 
